My potatoes are kept in the bag until I need to open the bag to retrieve them (usually a day or 2 after I buy them, b/c we eat potatoes frequently in a variety of forms). When I do this, I sort through them to see how many icky ones I have -- usually they have gashes from the harvesting or a bad bruise; occassionally I get one that is moldy -- and those are separated so I can use them up quickly -- usually as mashed potatoes. The rest are put into a reusable grocery bag (a black one) and stored on an open shelf under my microwave.
We don't use onions up quickly enough so I have stopped buying them in the 2-3 lb bags. I just buy 2-3 at a time and keep them in a bowl on my counter. I keep dehydrated onion bits in with the spices and use those in soups instead of fresh onion or I'll rehydrate them and use them in meatloaf or something similar. A lot recipes will call for a whole onion, but I only use 1/2, so the other half is chopped up and stored in the fridge for a few days until I need another half. If the onion is really large, I'll use 1/3 or 1/4 of it.